Special Session 145: Dynamic Models under Uncertainty in Economics and Finance

Regular stochastic flow and Dynamic Programming Principle for jump diffusions

Alessandro Bondi Luiss University Rome Italy Co-Author(s):

Abstract: Given a Brownian motion $W$ and a stationary Poisson point process $p$ with values in ${\\\\\\\\\\\\\\\\mathbb R}^d$, we prove a Dynamic Programming Principle (DPP) in a strong formulation for a stochastic control problem involving controlled SDEs of the form \begin{align*} \\\\\\\\ dX_{t}=&\\\\\\\\\\\\\\\\,b(t, X_{t}, a_t) dt + \\\\\\\\\\\\\\\\alpha \\\\\\\\\\\\\\\\left(t, X_{t}, a_t \\\\\\\\\\\\\\\\right) dW_t+\\\\\\\\\\\\\\\\int_{ \\\\\\\\\\\\\\\\{ |z| \\\\\\\\\\\\\\\\le 1 \\\\\\\\\\\\\\\\} } g\\\\\\\\\\\\\\\\left(X_{t-},t,z, a_t \\\\\\\\\\\\\\\\right)\\\\\\\\\\\\\\\\widetilde{N}_p\\\\\\\\\\\\\\\\left(dt,dz\\\\\\\\\\\\\\\\right) \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\ & + \\\\\\\\\\\\\\\\int_{ \\\\\\\\\\\\\\\\{ |z| >1 \\\\\\\\\\\\\\\\} } f\\\\\\\\\\\\\\\\left(X_{t-},t,z, a_t \\\\\\\\\\\\\\\\right){N}_p\\\\\\\\\\\\\\\\left(dt,dz\\\\\\\\\\\\\\\\right), \\\\\\\\\\\\\\\\quad X_s=x\\\\\\\\\\\\\\\\in\\\\\\\\\\\\\\\\mathbb{R}^d,\\\\\\\\\\\\\\\\,0\\\\\\\\\\\\\\\\le s \\\\\\\\\\\\\\\\le t \\\\\\\\\\\\\\\\le T.\\\\\\\\r\\\\\\\\n\\\\\\\\\\\\\\\\end{align*} Here $N_p$ [resp., $\\\\\\\\\\\\\\\\widetilde{N}_p$] is the Poisson [resp., compensated Poisson] random measure associated with $p$. We consider arbitrary predictable controls $a \\\\\\\\\\\\\\\\in {\\\\\\\\\\\\\\\\mathcal P}_T$ with values in any Borel set $B \\\\\\\\\\\\\\\\subset \\\\\\\\\\\\\\\\R^{l}$. The coefficients {$b$, $\\\\\\\\\\\\\\\\alpha$, and $g$} satisfy linear growth and Lipschitz--type conditions in the $x-$variable, and are continuous in the control variable. Moreover, we consider the value function $v(s,x)=\\\\\\\\\\\\\\\\sup_{a \\\\\\\\\\\\\\\\in {\\\\\\\\\\\\\\\\mathcal P}_T} , \\\\\\\\\\\\\\\\mathbb{E}\\\\\\\\\\\\\\\\big[\\\\\\\\\\\\\\\\int_{s}^{T}\\\\\\\\\\\\\\\\!h\\\\\\\\\\\\\\\\left(r,X_r^{s,x,a}, a_r\\\\\\\\\\\\\\\\right)dr + j\\\\\\\\\\\\\\\\left(X_T^{s,x,a}\\\\\\\\\\\\\\\\right)\\\\\\\\\\\\\\\\big] ,$ where $h$ and $j$ are suitable given maps, globally bounded and continuous in the $x-$variable and in the control variable. The DPP is a fundamental tool in stochastic control with applications in physics and mathematical finance. To prove it, we show the existence of a regular stochastic flow for the previous stochastic equation when the coefficients are independent of the control $a$. Notably, this regularity result is new for jump diffusions even when there is no large--jumps component, i.e., $f\\\\\\\\\\\\\\\\equiv0$ (cf. Kunita`s recent book on stochastic flows and jump diffusions). The proof of the DPP is completed by introducing an approach that relies on a new subclass of controls in $\\\\\\\\\\\\\\\\mathcal{P}_T.$ These controls allow us to apply a basic measurable selection theorem by L. D. Brown and R. Purves.

Stationary Mean-Field singular control of an Ornstein-Uhlenbeck process

Federico Cannerozzi Bielefeld University Germany Co-Author(s):

Abstract: Motivated by continuous-time optimal inventory management, we study a class of stationary mean-field control problems with singular controls. The dynamics are modeled by a mean-reverting Ornstein-Uhlenbeck process, and the performance criterion is given by a quadratic long-time average expected cost functional. The mean-field dependence is through the stationary mean of the controlled process itself, which enters the ergodic cost functional. We characterize the solution to the stationary mean-field control problem in terms of the equilibria of an associated stationary mean-field game, showing that solutions of the control problem are in bijection with the equilibria of this mean-field game. Finally, we solve the stationary mean-field game explicitly, thereby providing a solution to the original stationary mean-field control problem.

Turnpike properties in N-player differential games

Asaf Cohen University of Michigan USA Co-Author(s):    Jiamin Jian

Abstract: We consider the long-time behavior of equilibrium strategies and state trajectories in a linear quadratic $N$-player game with Gaussian initial data. By comparing the finite-horizon game with its ergodic counterpart, we establish exponential convergence estimates between the solutions of the finite-horizon generalized Riccati system and the associated algebraic system arising in the ergodic setting. Building on these results, we prove the convergence of the time-averaged value function and derive a turnpike property for the equilibrium pairs of each player. Importantly, our approach avoids reliance on the mean field game limiting model, allowing for a fully uniform analysis with respect to the number of players $N$. As a result, we further establish a uniform turnpike property for the equilibrium pairs between the finite-horizon and ergodic games with $N$ players. Numerical experiments are also provided to illustrate and support the theoretical results.

Focus or Diversification? Dynamic Allocation of Climate Technology Investments under Budget Constraints

Katia Colaneri University of Rome Tor Vergata Italy Co-Author(s):    Katia Colaneri, Alessio D`Amato and Ruediger Frey

Abstract: Major pathways for carbon abatement include a large-scale deployment of renewable energy sources (RES) and investment in carbon capture and storage (CCS) technologies. While RES such as solar and wind power offer clean, sustainable energy, significantly expanding their share in the energy mix necessitates heavy infrastructure investment. This is primarily due to issues of intermittency and the need to upgrade or redesign existing electricity grids to ensure stability and reliability. On the other hand, CCS technologies offer a potential solution to decarbonize existing fossil fuel-based infrastructure. However, CCS remains technologically immature and economically un-viable at large scale. Significant research and development efforts are required to reach a breakthrough that would make CCS a competitive option. Given limited fiscal capacity, it may be infeasible for societies to simultaneously invest heavily in RES infrastructure and fund foundational CCS research. This paper explores this trade-off by modeling the problem as a dynamic control problem. We analyze the optimal allocation of a constrained research and investment budget over time, under uncertainty about technological breakthroughs and deployment costs. A distinctive feature of our model is the lag between investment and technological availability. Our results highlight conditions under which strategic focus - favoring one technology over another - is optimal, versus cases where technological neutrality, i.e., parallel investments in both RES and CCS, may be justified.

Cooperation, Competition, and Common Pool Resources in Mean Field Games and Extensions with Inverse Learning

Gokce Dayanikli University of Illinois Urbana-Champaign USA Co-Author(s):

Abstract: The tragedy of the commons (TOTC) states that individual incentives result in the overuse of common pool resources (CPRs), which may have detrimental future consequences for everyone. However, in many real-life situations this does not occur, and researchers such as Nobel laureate Elinor Ostrom have suggested that mutual restraint by individuals can prevent it. In mean field games (MFGs), since individuals are insignificant and non-cooperative, TOTC is inevitable. This suggests that MFGs with CPRs must incorporate both selfishness and altruism to better capture real-world behavior. Motivated by this, we discuss equilibrium notions blending cooperative and non-cooperative actions. We introduce mixed-individual and mixed-population MFGs, together with modeling of CPRs. The former captures altruism at individual levels, while the latter represents a population composed of cooperative and non-cooperative individuals. For both, we outline equilibrium definitions, and their characterization via FBSDEs. We then present a fisheries-inspired example, discussing existence, uniqueness, and experimental results. Finally, we address the challenge that understanding intervention policies in mixed MFGs requires knowledge of individuals` altruism levels, which are unobservable. We therefore consider how these levels can be learned from data using inverse learning. (Based on works with Mathieu Lauriere, and with Xiaofei Shi, Haoyang Cao)

Existence of Strong Randomized Equilibria in Mean-Field Games of Optimal Stopping with Common Noise

Anna Pajola Bielefeld University Germany Co-Author(s):    Giorgio Ferrari

Abstract: We study a mean-field game of optimal stopping with common noise and investigate the existence of strong solutions via a connection with the Bank-El Karoui`s representation problem. Under certain continuity assumptions, where the common noise is generated by a countable partition, we show that a strong randomized mean-field equilibrium exists, in which the mean-field interaction term is adapted to the common noise and the stopping time is randomized. Furthermore, under suitable monotonicity assumptions and for a general common noise, we provide a comparative statics analysis of the set of strong mean-field equilibria with strict equilibrium stopping times. Based on a joint work with Giorgio Ferrari.

Optimal Regulation of Exhaustible Resources in a Mean-Field Model with Aggregate Uncertainty

Alexandros Pavlis LSE England Co-Author(s):    K. Kardaras, A. Pavlis N. Rodosthenous

Abstract: We develop a continuous time mean field model of an exhaustible resource extracted by a continuum of heterogeneous price taking firms under aggregate uncertainty, where the market price is determined endogenously by the aggregate outcome and a regulator sets a time dependent tax or subsidy. We formulate the regulator`s problem as a stochastic control problem over the evolving distribution of extractors, prove existence and uniqueness of an optimal policy, and establish qualitative properties of the regulator`s value function and optimal control. We then provide comparative statics for key primitives and complement the theory with numerical results illustrating equilibrium dynamics and the optimal regulatory policy.

An Optimal Energy Production Problem with Energy Source Switching and Load Following Nuclear Power Plants

Athena Picarelli University of Verona Italy Co-Author(s):

Abstract: The paper deals with a optimal energy production problem where the producer aims to match at any time a prescribed demand managing two types of energy sources: a renewable energy source subject to randomness and seasonality on one hand, and a controllable deterministic system of production (e.g. nuclear) on the other hand. The problem is formulated as a optimal switching problem over three possible regimes representing the increasing/decreasing/constancy of the nuclear production rate

Optimal Consumption and Portfolio Choice with No-Borrowing Constraint in the Kim-Omberg Model: The Complete Market Case

Tim Niclas Sch\"utz Bielefeld University Germany Co-Author(s):    Giorgio Ferrari

Abstract: In this paper, we study an intertemporal utility maximization problem in which an investor chooses consumption and portfolio strategies in the presence of a stochastic factor and a no-borrowing constraint. In the spirit of the Kim-Omberg model, the stochastic factor represents the expected excess return of the risky asset. It is perfectly negatively correlated with shocks to the risky asset and follows an Ornstein-Uhlenbeck process, thereby capturing the mean reversion of expected excess returns--a feature well supported by empirical evidence in financial markets. The investor seeks to maximize expected utility from consumption, subject to the constraint that wealth remains nonnegative at all times. To address the dynamic no-borrowing constraint, we use Lagrange duality to transform the primal problem into a singular control problem in the dual space. We then characterize the solution to the dual singular control problem via an auxiliary two-dimensional optimal stopping problem featuring stochastic volatility, and subsequently retrieve the primal value function as well as the optimal portfolio and consumption plans. Finally, a numerical study is conducted to derive economic and financial implications.

Irreversible reinsurance: Minimization of Capital Injections in Presence of a Fixed Cost

Maria Laura Torrente University of Genoa Italy Co-Author(s):    Salvatore Federico, Giorgio Ferrari

Abstract: We propose a model in which, in exchange to the payment of a fixed transaction cost, an insurance company can choose the retention level as well as the time at which subscribing a perpetual reinsurance contract. The surplus process of the insurance company evolves according to the diffusive approximation of the Cram\`er-Lundberg model, claims arrive at a fixed constant rate, and the distribution of their sizes is general. Furthermore, we do not specify any particular functional form of the retention level. The aim of the company is to take actions in order to minimize the sum of the expected value of the total discounted flow of capital injections needed to avoid bankruptcy and of the fixed activation cost of the reinsurance contract. We provide an explicit solution to this problem, which involves the resolution of a static nonlinear optimization problem and of an optimal stopping problem for a reflected diffusion. We then illustrate the theoretical results in the case of proportional and excess-of-loss reinsurance, by providing a numerical study of the dependency of the optimal solution with respect to the model`s parameters.

Learning Algorithm for Mean-Field Coarse Correlated Equilibrium: A Linear Programming Approach

Ioannis Tzouanas Bielefeld University Germany Co-Author(s):    Luciano Campi, Federico Cannerozzi

Abstract: We investigate the approximation of Coarse Correlated Equilibrium (CCE) within the framework of continuous-time mean-field games. In this setting, a regulator (or a correlation device) recommends strategies that agents have no unilateral incentive to deviate from. We begin by introducing the concept of optimal CCE and reformulating the problem using a linear programming approach to demonstrate existence under weak assumptions. Then, we focus on the approximation of these equilibria, we propose a novel no-regret primal-dual learning algorithm and prove its convergence. Finally, we provide numerical examples to illustrate our results.